A full understanding of Hartree-Fock theory would require a few weeks of
study. Here, we will aim for only a rough idea of what is going on.
Actually, you already know a few things about Hartree-Fock theory, even if
you haven't heard the term before. You almost certainly have heard about
molecular orbital (MO) theory, or the linear combination of atomic orbitals
molecular orbital (LCAO-MO) theory. This is actually the same as
Hartree-Fock theory, which is named after the two physicists who invented
it. The basic idea is that we will describe the motion of each electron by
a molecular orbital. The mathematics behind it is that each MO is made of a
linear combination of atom-centered basis functions. The Hartree-Fock
procedure simply solves for what the linear expansion coefficients actually
are.

The variables in the Hartree-Fock equations unfortunately depend on
themselves, so they must be solved in an iterative manner. You will see
these iterations in the outputs you run in the lab. In typical cases, the
Hartree-Fock solutions can be obtained in roughly 10 iterations. For tricky
cases, convergence may be improved by changing the form of the initial
guess. Since the equations are solved self-consistently, Hartree-Fock is an
example of a self-consistent field (SCF) method.

Unfortunately, you are probably under the illusion that molecular orbitals
are somehow ``real'' or ``true.'' Except for the special case of the
hydrogen atom, this is completely false. Molecular orbitals are the
product of Hartree-Fock theory, and Hartree-Fock is not
an exact theory: it is an approximation to the electronic Schrödinger
equation. The approximation is that we pretend that each electron feels
only the average Coulomb repulsion of all the other electrons. This
approximation makes Hartree-Fock theory much simpler than the real
problem, which is an N-body problem. Unfortunately, in many cases this
approximation is rather serious and can give bad answers. It can be
corrected by explicitly accounting for electron correlation by density
functional theory (DFT), many-body perturbation theory (MBPT), configuration
interaction (CI), and other means.